#6506412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25496	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(17,18,22,24,20,23,21,26)(19,27,30,32,28,25,29,31);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(24,26)(29,30);(9,11)(10,13)(12,16)(14,15)(18,24)(21,22)(23,26)(25,31)(27,32)(29,30);(11,16)(13,15)(17,19)(18,25)(20,28)(21,29)(22,30)(23,27)(24,31)(26,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,23)(19,28)(24,26)(29,30)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413608);